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LUMPED PARAMETER ANALYSES OF STREAM HYDROGRAPHS
A stream hydrograph is the electrocardiogram of a watershed, recording the
hydrologic pulse of the region as it is stressed by drought or flood. It is no wonder that the
analysis of stream hydrographs has fascinated hydrologists as a method for diagnosing
processes within the watershed drained by the stream. This interest began more than a
century ago with the work of Boussinesq in the late 1800s, and it continues today with both
applications to field studies and the development of new methods.
The flow in a stream is generally recognized to have contributions from ground
water and both overland flow and shallow subsurface stormflow. Overland and stormflow
typically change over time periods of hours or days following precipitation events or
snowmelts, whereas baseflow changes gradually over many days or weeks in response to
seasonal or climatic variations, or to the drainage of water from an aquifer of finite volume.
In some watersheds, drainage from the unsaturated zone may be also be an important
contributor to the long-term discharge of streams (Hewitt ).
The basic process of analyzing a hydrograph begins by separating the components
of stormflow and baseflow. This is done to facilitate analysis, but it also arises out of the
different purposes that hydrograph analysis may serve. Investigators of flooding are
primarily interested in the stormflow component, whereas investigators of ground water or
long-term effects of water fluxes are more interested in the baseflow component. The
separation process has received considerable attention but the first step of most of the
proposed methods give similar results. This initial step is to go through the hydrograph
record and identify days when the flow in the stream is primarily due to baseflow
contributions. Such days are baseflow turning points. Typically, daily flows are only
accepted as indicative of baseflow conditions if they occur during a period when the flow is
gradually decreasing. Some methods use sophisticated filtering methods, whereas others
use graphical or simple numerical procedures. They all identify turning points as days of
low flows between pulses of high flow caused by storms, and most of them will pick
similar days if implemented with appropriate parameters. Baseflow turning points
identified by a method recommended by the Institute of Hydrology in England are shown
in Figure 2.
The times identified as baseflow turning points may be days or weeks apart,
particularly during a rainy season. In between the turning points the stream is flowing with
contributions from both baseflow and stormflow. A complete hydrograph analysis requires
baseflow and stormflow components to be identified throughout the record, so some
method of interpolating between turning points is required. A variety of methods have
been proposed, and herein lie the major differences between baseflow separation methods.
The baseflow component of many hydrographs has a distinctive behavior during
dry weather. Notice how the minimal flows decrease during the summer months of the
three years plotted on Figure 1. When plotted on semi-log axes, the minimum flows form a
roughly straight line with a negative slope during the summer months. The baseflow
component is not shown on the figure, but it corresponds to the minimal flow on the
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hydrograph. As a result, the baseflow component appears to decrease as a negative
exponential during the summer months. In South Carolina, the summer months commonly
have relatively small amounts of rainfall and evapotranspiration captures much of the
rainfall that does occur. As a result, we
expect that recharge to the aquifer that
discharges to the Little River was
2500
negligible during summer months times
2000
when the baseflow decreased as a negative
exponential. These periods of decreasing
1500
baseflow are called baseflow recessions.
1000
We infer that baseflow recessions occur
when the recharge is negligible and the
500
aquifer is continuously discharging to the
0
stream.
1600
Discharge (cfs)
1400
1200
1000
800
600
400
200
Rainfall (inch/month x 100)
1800
0
J FMAM J J A SON
J FMAM J J A SON
J FMAM J J A SON
Discharge (cfs)
1600
1400
e7
1200
1000
e6
800
600
e5
400
200
e4
Rainfall (inch/month x 100)
1800
e8
0
J FMAM J J A SON
J FMAM J J A SON
J FMAM J J A SON
Figure 1. Hydrographs and hyetographs
from the Little River, SC. for three years.
ln (discharge (ft^3/s))
5
4
3
1-Jan-88
1-Apr-88
1-Jul-88
30-Sep-88
30-Dec-88
Figure 2. Hydrograph with baseflow
turning points selected using a simple
filtering algorithm (open circles).
Baseflow recessions are important
to the management of both ground water
and surface water resources during
drought. The continuously diminishing
baseflow during a recession is responsible
for the diminishing stream flows during a
drought. Understanding the controls of
baseflow recession should help to
understand to amount and rate of water
that is available during a drought.
The objective of this analysis is to
develop a method for predicting the
baseflow to a stream as a function of time
and recharge, and hopefully to gain some
insight into baseflow recessions. This will
allow us to interpolate between baseflow
separation points, which will give
the baseflow and stormflow
components as continuous
functions of time. This modest
analysis can be extended to
provide a method for estimating
the major fluxes throughout a
watershed as a function of time,
which is a remarkable result.
CONCEPTUAL MODEL
The hydrogeology of nearly
any watershed is extremely
complicated when viewed in detail.
It is possible to assemble a 3-D
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model that considers the spatial variations in material properties and processes present in a
watershed, and considerable insights can be obtained from this type of model. These are
called “distributed parameter” models because values of parameters, like hydraulic
conductivity, are distributed and can vary across the model. Distributed parameter models
can resemble the geometry of a watershed, and they can provide valuable insights when
geometry is important. However, distributed models are time consuming to build and
calibrate, and so the insights provided from such a model must be worth the effort of
putting it together.
An alternative approach is to idealize the watershed as a simplified system that can
be characterized using a small number of parameters. The many distributed parameters of
the 3-D model are thereby lumped together to form one, or a few, effective parameters in a
simplified model whose behavior resembles the actual watershed. This type of simplified
analysis is called a “lumped parameter” model. Many of the models that we have
described in the previous problems are types of lumped parameter models.
R(t)


h
b
A’
x
L
A
A
A’
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We will idealize a watershed as a sand-filled tank that is drained by a sand-filled
hose held at constant head. This ignores the pattern of streams in the watershed, however,
the tank analogy is a simplified approximation of the in the cross section in Figure 2; that
is; the tank and sand-filled hose resembles the aquifer that discharges to a nearby stream.
The discharge from the hose will decrease as the water level in the tank falls. We will
equate the discharge from the hose to the contribution of baseflow to the discharge in a
stream. The tank analogy can be justified on more theoretical grounds because the
analytical solution for the problem shown in the cross-section consists of an infinite series.
The first, and largest, term in the series is the same expression that we will derive for the
tank. The higher order terms in the series account for the geometry of the aquifer.
The tank is a model of an aquifer with inputs from recharge and outputs from
baseflow. The water stored in the aquifer changes when the water level changes.
Figure 2. Plan and section views of a simple watershed.
R
Aw
L
h
baseflow
Figure 3. Tank analogy for a watershed.
The objective of the analysis is to develop expressions for the average head in an
aquifer and the baseflow discharge to a stream as a function of time and recharge. We
want to use this analysis to understand baseflow recessions, and to predict the baseflow
between any pair of baseflow turning points on a hydrograph.
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Translation
1. Rate In The rate of water flowing into the aquifer is
Qin = RAw
(1)
Where Aw is the area of the watershed in plan view.
2. Rate Out
The rate being discharged by baseflow is
Qout = qoutAa
(2)
Where Aw is the area of the aquifer along a cross-section parallel to the stream. From
Darcy’s Law the baseflow flux is
qout = K
h
L
(3)
so
Qout =
KAa
h
L
(4)
3. Rate of change of storage. Water stored in the aquifer is
Vw  nAw h
(5)
where n is the specific yield and the rate of change of storage is
dVw
dh
 nAw
dt
dt
(6)
Substituting (1) (4) and (6) into a volume balance gives
RAw 
KAa
dh
h  nAw
L
dt
(7)
rearranging
dh R KAa
 
h
dt n nAw L
Solution
Separating variables
(8)
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dh
 dt
R KAa

h
n nAw L
and integrating

nAw L  R KAa 
ln  
h   t  Cint
KAa  n nAw L 
(9)
We will assume that we know the water level at some time and are interested in the water
levels after that time, so the initial condition is that h = ho when t = to. This gives


nAw L  R KAa
ln  
ho   Cint
KAa  n nAw L 
(10)
and substituting into (9) gives
R 
 KAa
h 

nAw L
n 
KAa
ln 

t
 KAa
R
nAw L
ho  

n
 nAw L
(11)
Introducing the lumped parameter
a
KAa
nAw L
(12)
allows us to simplify
R 

 h

an

   at
ln
R 

 ho 

an 

R
R

h   ho  e  at 
an 
an

h
R
1  e at 
 e  at 
ho
aho n
(13)
This gives the head in the aquifer as a function of time. We can determine the baseflow by
substituting (4) into (13) to get ho = QoL/KAa and
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Q
h

Qo ho
(14)
so from (13)



A KR
Q
R
 e at 
1  e at  e at  a
1  e at
Qo
aho n
aLQo n

(15)
where Qbo is the baseflow at t = 0 and Qb is the baseflow after that time. We should
introduce another lumped parameter to characterize recharge
R* 
RAa K
A
R w
aLQo n
Qo
(16)
to get

Q  Qo e  at  RAw 1  e  at

(17)
or in dimensionless form
Q
 Q*  e t*  R * 1  e t* 
Qo
(18)
Baseflow is given as a function of time and recharge rate by (17) or (18). The
response of the system is govern by the value of the watershed recession constant a. We
probably should not take the geometric implication of a too literally—clearly the geometry
of the aquifer will differ markedly from that of a tank and a hose. Nevertheless, a
meaningful interpretation is to consider a as composed of terms related to geometry and
terms related to hydrologic properties of the aquifer. This is important because we see that
the expression in brackets depends only on the physical characteristics of the aquifer.
Often this expression is simplified to a single value, the recession constant (Fetter, 2001).
The volume of water that is stored, or that can be released during a recession can be
estimated from the analysis above. This type of estimate is important because it allows the
volumetric flowrate during a drought to be forecasted. Moreover, the difference between
the volume that is stored at the beginning of one recession compared to that from the
previous recession is the volume that has been recharged to the aquifer.
Assuming the recharge is zero during a recession then (17) reduces to the baseflow
during a recession
Q  Qo e  at
(18)
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Note that the analysis predicts that the baseflow during a recession decreases as a negative
exponential of time.
Calibration
The analysis of baseflow as a function of time can be calibrated by determining a value of a
that best predicts hydrograph data from a particular watershed. Perhaps the easiest way to
approach this is to determine the semi-log slopes of the recession portions of hydrographs
during baseflow recessions. We assume that the recharge is negligible during a recession,
so the semi-log slope should be equal to a. In practice the semi-log slopes of different
baseflow recessions will vary somewhat, probably because some recharge occurs during
recession. The usual practice is to measure the slopes from many recession events and
determine the semi-log slope that either best represents these values. Some hydrologists
recommend using an average slope, whereas others (like me) prefer that a should be
slightly less than the greatest observed slope. In any case, we will assume that a can be
obtained from hydrograph data describing recessions.
Volume of water in storage
The volume of water that could be released from storage in an aquifer will have
applications in managing a water supply during drought. We can obtain an estimate for the
volume of water that has been discharged from the aquifer from t = 0, the start of the
recession, to t = t1 is obtained by integrating (18)
t1

V (t1 )  Qo  e at dt  
0
Qo at
e
a
t1
0

Qo

(1  e at1 )
a
(19)
The maximum volume of water that can be released during the recession occurs when t1=.
Vmax 
Qo
a
(20)
so
V (t1 )  Vmax (1  e  at1 )
(21)
It may be convenient to recast (19) as
V (t ) 
Qo
Q
Q( t )
1
(1  e  at )  o (1 
)  (Qo  Q(t ))
a
a
Qo
a
(22)
thereby giving the volume of water released by baseflow at time t in terms of the difference
between the initial discharge and the discharge at t. The volume stored in the aquifer that
could be released is simply the difference between Vmax and V(t), or
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Vstored (t ) 
Q (t )
a
(23)
This shows a simple relation between the volume of water stored in an aquifer, the
baseflow, and the recession constant.
Recharge
The recharge rate between any pair of turning points can be estimated by
rewriting (17) so that Q = Q1 is the baseflow that occurs at t = t1

Q(t )  Q1e  a (t t1 )  RAw 1  e  a (t t1 )

(24)
Similarly, Q = Q2 is the baseflow that occurs at t = t2. Recharge to the aquifer that occurs
from t1 to t2 can be determined by rearranging (24)
R1, 2
Q2  Q1e  a (t2 t1 )

Aw 1  e a (t2 t1 )


(25)
This simple analysis gives the average recharge flux in an aquifer between the times
indicated by any pair of turning points, say on Figure 2. Applying (25) sequentially
along a hydrograph will allow the recharge to be determined with time.
A recharge pulse followed by a recession: superposition in time
The analysis given above shows how the baseflow contribution to a stream
changes during a recharge event of infinite duration, which is interesting but a bit
unrealistic. We can use a simple trick to transform this result to give the baseflow during
a recharge event where the recharge is constant and equal to R, followed by a recession
event where the recharge is zero. We will assume the recharge event begins at t = 0 and
ends at t = t1. During the recharge event, we simply use the original solution in (13).
After the recharge even ends, however, the approach is to superimpose two solutions, one
for a recharge event of magnitude R that begins at t=0 and is of infinite duration (the
solution developed above), and another for a recharge event of magnitude –R that starts
at t = t1 and continues indefinitely.
R
R
R
t1
t
t
t1
t
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Baseflow for t<t1
Qt t 1  Rs 1  e  at 
(26)
and it follows that the baseflow at the end of the recharge period is
QR  Rs 1  e  at1 
(27)
Baseflow for t>t1, during recession
Here we need to superimpose the two analyses. The baseflow due to a recharge
event of magnitude –R that starts at t=t1 is obtained directly from (27)
Q   Rs 1  e  ( t * t1 *) 
(28)
Adding (26) and (28) and doing some algebra
Qt t1  Rs 1  e  at   Rs 1  e  a ( t t1 ) 
Qt t1  Rs 1  e  at  1  e  a ( t t1 ) 
Qt t1  Rs  e  at  e at1 e  at   Rs e  at (e at1  1)
Now multiply by
e  at1
and rearrange to get
e  at1
Qt t1   Rs (1  e  at1 ) e  a ( t t1 )
but according to (27) the term in curly brackets is Qr, the discharge at the start of the
recession.
The discharge during a recession that begins at t1 is
Qt t1  QR e  a ( t t1 )
(29)
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0.8
1.8
1.6
0.6
1.4
1.2
1
0.4
0.8
0.6
0.2
0.4
0.2
0
0.5
1
1.5
t
2
2.5
3
0
Figure 1. Q* as a function of t*
for R*=0.5, 1 and 2.
1
2
t
3
4
Figure 2. Discharge during a
recharge event during t*<1
followed by recession. Based on
(30).
Multiple recharge events of various magnitude
Clearly the recharge rate in a basin varies as a function of time. In general,
recharge will be equal to precipitation minus interflow and evapotranspiration. We
expect that the recharge will probably vary through the year and roughly follow the same
trends as rainfall. Often rainfall is expressed as a hyetograph, where the total rainfall
over a period is given as an average rate over that period. Thus, we can idealize recharge
as a sequence of events of some duration and average magnitude. The analysis above
gives the discharge resulting from one such event. We will extend that analysis to
consider any number of events.
The flux at the end of the first recharge period, t = t2, is
q2  R1[1  e  a ( t2 t1 ) ]  q1e  a ( t2 t1 )
(31)
The recharge during period 1 has been labeled R1.
It follows that the flux during period 2, from t2 < t <t3 is
q  R2 [1  e  a ( t t2 ) ]  q2 e  a ( t t2 )
(32)
but we can substitute (31) to give
q  R2 [1  e  a ( t t2 ) ]   R1[1  e  a ( t2 t1 ) ]  q1e  a ( t2 t1 ) e  a ( t t2 )
q  R2 [1  e a ( t  t 2 ) ]  R1[e a ( t  t 2 )  e a ( t  t1 ) ]  q1e a ( t  t1 )
(33)
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Including another recharge interval
q  R3[1  e a ( t  t 3 ) ]  q3e a ( t  t 3 )
but from (33), the discharge at the end of the second interval (starting the third interval) is
q3  R2 [1  e a ( t 3  t 2 ) ]  R1[e a ( t 3  t 2 )  e a ( t 3  t1 ) ]  q1e a ( t 3  t1 )
Substituting
q  R3[1  e a ( t  t 3 ) ]  R2 [1  e a ( t 3  t 2 ) ]  R1[e a ( t 3  t 2 )  e a ( t 3  t1 ) ]  q1e a ( t 3  t1 ) e  a ( t  t 3 )
and simplifying
q  R3 [1  e a(t t3 ) ]  R2 [e a(t t3 ) ]  e a (t t2 ) ]  R1[e a (t t2 )  e a (t t1 ) ]  q1e a(t t1 )
It is apparent that the flux can be determine after any number of recharge events simply
by adding more terms as in (33). This concept can be implemented as
m 1
q (t )  q1e  a ( t  t2 )  Rm [1  e  a ( t  tm ]   Ri [e  a ( t  ti 1 )  e  a ( t  ti ) ]
(34)
i 1
The first term describes the decay of the initial flow rate as if the recharge that follows is
zero. The summation describes the effects of subsequent recharge events of varying
magnitude (possibly = 0). The magnitude of the ith recharge event is Ri, and the duration
is from ti to ti+1. Eq. (34) assumes that t  tm.
Getting fancier
The analysis above is unable to predict the actual discharge of the stream because
it ignores the substantial contribution of interflow or overland flow following rainfall.
We could substantially improve our model by including the processes of shallow
saturated flow. This would allow us to include the interflow component, and it would
also allow us to calculate the recharge using records of rainfall and evapotranspiration,
rather than estimating it empirically. A simple water balance on the vadose zone
indicates
Precipitation = interflow + recharge + change in storage + evapotranspiration
P = I + R + S + ET
(35)
The approach will be to assume that there are two water reservoirs, one shallow
and governed by the water balance above, and a deep groundwater system that behaves
according to (34). We will use the CSR analogy for the shallow system and develop the
analysis the same way that we did for the deep one. The two systems will be coupled
together through the recharge. Interflow will be assumed to behave the same as baseflow
for the groundwater system. We must use different constants, one for the deep system,
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ad, and another for the shallow system, as. Similarly, we will assume that the two
systems are characterized by different head, hs and hd. Substituting as before, we change
the water balance based on average flux (35) to one based on volumetric flow
At s P 
KAps
dh
hs  R  nAt s s  At s ET
x
dt
(37)
Rainfall
ET
Recharge
hs
HR
h
Interflow
b
aseflo
w
x
hd
baseflow
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We will assume that recharge occurs after the head in the shallow system reaches some
threshold value, HR. No recharge occurs if hs<HR. Recharge flux will be proportional to
(hs - HR) when hs>HR. The volume of water in the tank below HR is the water that can be
stored in the vadose zone against gravity drainage.
So
R0
KApR
R
( h  HR )
x s
I 0
I
hs<HR
hs>HR
hs <H I
KApI
x
(hs  H I )
hs >H I
(38)
Using an similar threshold concept for interflow, and substituting
0
KApI
x
(hs  HI ) 
KApR
x
(hs  HR )  Sy At s
dhs
 ( ET  P) Ats
dt
(39)
solving for the derivative
KApI
KApR
dhs
( ET  P)

(hs  H I ) 
(hs  H R ) 
dt
Sy At s x
Sy At s x
Sy
and grouping terms gives
(40)
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dhs
a
a
  s1 (hs  H I )  s 2 (hs  H R )  F
dt
Sy At s
Sy At s
KAps
as1 
hs  H I
x
as1  0
as 2 
hs<H I
KApR
hs>H R
x
as 2  0
F
(41)
hs<H R
( ET  P)
Sy
Simplifying further
dhs
  Ahs  B
dt
A  (a s1  a s 2 )
B  a s1 H I  a s 2 H R 
(42)
( ET  P)
n
Note that the interflow response to rainfall is expected to lag behind the actual
precipitation recorded at rain gauges due to the travel time from the point where the rain
lands in the basin to the point where it is measured at the gauging station. This delay will
be included in this model by introducing a lag time, tL, to the precipation record.
Accordingly,
F(t) = P(t-tL) + ET (t-tL)
Where for completeness evapotranspiration is also assume to lag by tL. In any
case, it will be assumed that F(t) is constant during a particular time interval.
Eq. (42) is separable and the solution is
t
1
ln(  Ahs  B)  Cint
A
(42)
1
ln(  Ahs1  B)
A
(43)
hs = hs1 when t = t1
Cint  t1 
t
1
1
ln(  Ahs  B)  t1  ln(  Ahs1  B)
A
A
(44)
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 At  t1   ln(
B  Ahs
)
B  Ahs1
B  Ahs   B  Ahs1 e A t  t1 
hs 
B B

   hs1  e  A t  t1 


A
A
(45)
(46)
(47)
The head in the upper reservoir after time t1 is
hs 
hs 



B
1  e  A t  t1   hs1e  A t  t1 
A
(48)

a s1 H I  a s 2 H R  RET1
1  e  ( as1  as 2 )  t  t1   hs1e  ( as1  as 2 )  t  t1 
a s1  a s 2
Application
This analysis predicts the effective head in the shallow system as a function of
precipitation and ET and four empirical constant; as1 and as2, are rate constants for the
drainage of water from the reservoir as recharge or interflow, and HI and HR, are
threshold heads for the initiation of interflow and recharge. The area of the basin and the
specific yield are also required.
The approach is to determine RET from available data. Then use (48) to
determine hs. Interflow is determined using I=aIhs, where aI is another constant.
Recharge is determined after hs has been found. It is obtained using (38). Baseflow is
determined using (30). The total discharge of the stream is Baseflow + Interflow.
The analysis is now a function of five constants, as1, as2, aI, HR and ab. aI and As2
are related through the basin area and porosity. This gives a total of 4 constants. Use the
record of stream discharge, rainfall and ET to determine constants. Then use eqs. above
to determine the baseflow, interflow, recharge and change of water stored in vadose zone
as functions of time.
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CU HYDROGRAPH
CU Hydrograph is a program that implements the analysis outlined above, along
with some other analyses to interpret stream hydrographs. The program does hydrograph
separation to identify turning points and then it uses several different models for
estimating baseflow and recharge from the turning points. At the end of this analysis you
have Precipitation, Baseflow, Recharge, Total Streamflow, Stormflow, and Rate of
change of storage in the aquifer. ET is estimated using the balance on the vadose zone:
ET= Precip- Stormflow-Recharge
This completes the major components of the water balance for a watershed. You need
data describing total stream flow and precipitation, and the analysis produces an estimate
of all the transient fluxes in the watershed.
You need to enter flow data from a stream gauging station, and precipitation data
to start the analysis. You will also need to enter the correct ranges for the data. Do this
by going to Insert-Name-Define and editing the appropriate variables. These variable tell
the program where to find the data.
Baseflow separation and Segtime Go to the Hydrograph 1 worksheet and do a baseflow
separation analysis to distinguish baseflow from total flow. The method described by the
Wallingford Institute of Technology is currently implemented in the program, although
another slightly different method can be implemented with a small change in the code.
Using the baseflow separation routine requires that you specify a segment time, which is
the time in days required for the effects of a storm pulse to decay. Pressing the button
called “Separate by Institute of Hydrology Method” will divide the hydrograph into
segments of length=segment time. Then it determines the lowest flow in each segment.
Then it compares each low flow to the low flows in each of the neighboring segments.
The idea is that baseflow changes fairly slowly, so the change between one low flow and
the neighboring lows will be fairly small if the point represents baseflow. This is
implemented by taking 0.9 of the current point and comparing this value to the
neighboring lows. The point is recognized as baseflow if the 0.9 value is less than both
neighbors. This filtering mechanism seems somewhat arbitrary—the justification for 0.9
is unclear for example. However, the method seems to give reasonable results.
Another method can be implemented where baseflow is identified by the slopes between
the current point and both the previous and the following points. These slopes are
specified by maxup and maxdown on the Hydrograph 1 worksheet. To use this method
you need to go into the code and call FilterbySlope instead of FilterbyInstHydro. I tried
the FilterbySlope algorithm and it gives results that are similar to FilterbyInstHydro.
The program will display all the points that were selected as minimum values in each
segment, and it will identify which of those minimum values is baseflow. The points
identified as baseflow are called turning points. Adjust the value of segtime and
recalculate the baseflow turning points. You need to decide whether the points that are
being selected are a reasonable representation of baseflow.
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Note that you may need to adjust the ranges on the graph for it to display all, or even part,
of the data.
Basin Constant The analysis requires selecting a basin constant, called a in the analysis
above. The basin constant is a lumped parameter that includes effects related to the size
and properties of the watershed. This constant must be determined by the user and there
is a worksheet for assisting in this process. The basin constant will be the semi-log slope
of the baseflow hydrograph during recession; that is, when the recharge is zero. Many
hydrographs have straight sections with a negative slope in dry months, and this is
interpreted as recession. The basin constant is determined by obtaining the slope of the
line representing recessions.
The basin constant will have a significant effect on the results, so it is important to select
an appropriate value. I think the most appropriate slope to select here is one that is either
equal to, or nearly equal to the steepest negative slope during recession. The reasoning is
that any slope that is more positive than that represented by the basin constant will be
interpreted by the program as recharge. Picking a that is too small will produce negative
recharge, which is inconsistent with the model.
The Basin Constant worksheet has space to construct a Master Recession Curve, and it
automatically gives some statistics that can be useful in selecting a. A Master Recession
Curve is constructed by drawing straight lines representing recessions on a semi-ln
hydrograph (this is automatically created in the example). Slide the lines together and
determine the slope of the steepest, or nearly the steepest one. Be certain the units are
correct. This is one estimate for the Basin Constant.
One of the plots gives values of slope determined using different numbers of turning
points. The slope steepens as the number of points increases. Another gives these same
data as probabilities.
Enter the value for the basin constant in Hydrograph1 worksheet.
Watershed Area Enter the basin area in ft2 in the Hydrograph1 worksheet.
Calculations
Before you do the calculations, you should be sure you have entered
1. Segment time
2. Basin Constant
3. Watershed area
Now go the the Results by Day worksheet. Press the button to do the calculations. The
screen will flash awhile and then the results will be displayed.
The plots on this screen are set up to display data from the examples. You probably will
have to change the range of the x axis to display your data.
The data on this page include the water fluxes estimated on a daily basis. The program
has calculated 10 years worth of data, although the display may only show a subset of
this information. You can change the amount of daily data written to the screen by
modifying the code.
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The analysis does calculations assuming that recharge occurs by four possible schedules.

Uniform rate between turning points

Instantaneous pulse at the first turning point.

Instantaneous pulse at the time of the peak in total flow

Sinusoidal distribution
Probably the actual schedule varies somewhat depending on the details of the
precipitation history, and how flow processes are averaged through the watershed. In
general, the uniform rate gives the least amount of total recharge and the instantaneous
pulse gives the greatest, although the differences are fairly minor. The program
interpolates the baseflow between the turning points, and the shape of this curve is
noticeably affected by which application schedule you choose.
Results by Month
Some applications require average values of flux, and these results are calculated in a
worksheet called Results by Month. The data include all the monthly averages for all the
fluxes. Monthly averages and total averages of fluxes are summarized in the colored
boxes at the bottom of the page.
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The following are some example sheets from the program.
Some instructions about how to use the program are on this worksheet.
Enter data here
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Then do the hydrograph separation
Be sure to enter the Basin Area, as well as the segment length and the Basin Constant.
Press the button to calculate the turning points. You may need to adjust the ranges on the
graph.
Use this worksheet to help determine a
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Then go here and press the button to calculate the daily fluxes.
A summary of the monthly and yearly fluxes is also calculated. These values could be
useful when calibrating numerical models.
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This spreadsheet will perform hydrograph separation analyses according to the class notes.
It requires input in the form of daily measurements of streamflow and monthly precipitation
Put those data in the data worksheet
Go to hydrograph 1 worksheet to do a hydrograph separation analysis and to determine
a basin constant a, along with the other parameters on the sheet
Press the button to revise the hydrograph.
Note that a plot of the slopes between each pair of baseflow points is given on the MRC worksheet
The program current uses the hydrograph separation method described by the Wallingford Institute of Hydrology
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There are graphs on the Basin Constant worksheet that may help you determine a
You must enter the time base (segment length), a, and the watershed area on the Hydrograph 1 worksheet
After you have determined the time base, a, then go the Results by day worksheet
Press the on the Results by day worksheet to calculate and display components of the water balance by day and by m
The button on Hydrograph 1 calls a subroutine called Separate to do the baseflow separation
The Button on Results by Day calls a subroutine called MainProgram to determine the water balance
All subroutines are in Module 1 and all functions are in Module 2 in the attached Vbasic file
Note that all the functions to calculate the components of the water balance are accessible and can be used for other
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This spreadsheet will perform hydrograph separation analyses according to the class
notes.
It requires input in the form of daily measurements of streamflow and monthly
precipitation
Put those data in the data worksheet, Be sure to revise the values representing the ranges of the
data
Go to hydrograph 1 worksheet to do a hydrograph separation analysis and to
determine
a basin constant a. You must include the area of the watershed on this spreadsheet
Press the button to revise the
hydrograph.
The program current uses the hydrograph separation method described by the Wallingford Institute of
Hydrology
There are graphs on the Basin Constant worksheet that may help you determine a
You must enter the time base (segment length), a, and the watershed area on the Hydrograph 1 worksheet
After you have determined the time base, a, then go the Results by day worksheet
Press the on the Results by day worksheet to calculate and display components of the water balance by day and by
month
The button on Hydrograph 1 calls a subroutine called Separate to do the baseflow
separation
The Button on Results by Day calls a subroutine called MainProgram to determine the water
balance
All subroutines are in Module 1 and all functions are in Module 2 in the attached
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Vbasic file
Note that all the functions to calculate the components of the water balance are accessible and can be used for other calculations